\catcode`"=\active
\gdef"#1"{{\sf #1}}

\newcommand{\id}[1]{{\it #1}}
\newcommand{\junk}[1]{}

%\setlength{\parskip}{0.5ex}
%\setlength{\parindent}{0pt}
\newcommand{\qed}{\hfill \rule{7pt}{7pt}}
\newcommand{\Eqnum}[1]{(\ref{eqn:#1})}
\newcommand{\Eqn}[1]{Equation~\Eqnum{#1}}
\newtheorem{Definition}{Definition}[section]
\newtheorem{Theorem}{Theorem}
\newtheorem{Remark}{Remark}
\newtheorem{Observation}{Observation}
\newtheorem{Fact}{Fact}
\newtheorem{Claim}{Claim}
\newtheorem{Lemma}{Lemma}[section]
\newtheorem{Corollary}{Corollary}[Theorem]
\newtheorem{Lcorol}{Corollary}[Lemma]
\newtheorem{Example}{Example}
\newcommand{\example}[1]{
\begin{Example}
{\em #1}\qed
\end{Example}
}
\newenvironment{Proof}{\noindent{\bf Proof: }}{\qed}
\newenvironment{LabeledProof}[1]{\noindent{\bf Proof of #1: }}{\qed}
\newcommand{\ProofSpace}{\newline}
\newenvironment{Sketch}{\noindent{\bf Proof Sketch: }}{\qed}
\newenvironment{LabeledSketch}[1]{\noindent{\bf Proof Sketch of #1: }}{\qed}
\newenvironment{Invariant}[1]{\begin{quote}{Invariant\em
#1\/}:}{\end{quote}}

\newcommand{\BulletSlPara}[1]{\noindent{$\bullet$ {\sl #1}}.}
\newcommand{\SlPara}[1]{\smallskip \noindent{\sl #1}.}
\newcommand{\BfPara}[1]{\noindent{\bf #1}.}
\newcommand{\ScPara}[1]{ \noindent{\sc #1}.}

%\newcommand{\SlPara}[1]{\noindent{\sl #1}.}

\newcommand{\equals}{\stackrel{\mbox{\tiny{def}}}{=}}
\newcommand{\ceil}[1]{\left\lceil  #1 \right\rceil}
\newcommand{\floor}[1]{\left\lfloor  #1 \right\rfloor}
\newcommand{\bin}[2]{\mathop{\mbox{\rm bin}}(#1,#2)}
%\newcommand{\smallbin}[2]{\!\!\mathop{\mbox{\scriptsize{\rm bin}}}(#1,#2)}
\newcommand{\smallbin}[2]{\mathop{\mbox{\scriptsize{\rm bin}}}(#1,#2)}
%\newcommand{\qed}{\protect\raisebox{.3ex}{\framebox[.5em]{\rule{0em}{.6ex}}}}
\newcommand{\dis}{\mathop{\mbox{\rm dist}}\nolimits}

\newcommand{\ihat}{\hat{\imath}}
\newcommand{\jhat}{\hat{\jmath}}
\newcommand{\eps}{\varepsilon}

\newcommand{\poly}{\mbox{poly}}
\newcommand{\polylog}{\mbox{polylog}}
\newcommand{\Real}{\mbox{\bf R}}
\newcommand{\Nat}{\mbox{\bf N}}
\newcommand{\Int}{\mbox{\bf Z}}

\newcommand{\crunchList}{\itemsep 0em}

\newtheorem{ResearchProblem}{Problem}
\newcommand{\Research}[1]{
        \begin{center}
        \fbox{
        \begin{minipage}{5.25in}
        \begin{ResearchProblem}
        #1
        \end{ResearchProblem}
        \end{minipage}
        }
        \end{center}
}

\newcommand{\Comment}[1]{{\bf Comment}: #1}

\newcommand{\Response}[1]{{\bf Response}: #1}

%%% Emphasis in definitions
\newcommand{\dem}[1]{{\bf #1}}

%\newcommand{\mod}{\mbox{ mod }}

\newcounter{listCounter}
\newenvironment{Enumerate}{\begin{list}{\arabic{listCounter}.}{
        \usecounter{listCounter}\ListLengths}}{\end{list}}
\newenvironment{AlphaList}{\begin{list}{(\alph{listCounter})}{
        \usecounter{listCounter}\ListLengths\setlength{\labelwidth}{3ex}}}{\end{list}}

\newenvironment{Itemize}{\begin{list}{$\bullet$}{\ListLengths}}{\end{list}}
\newenvironment{InnerItemize}{\begin{list}{$-$}{\ListLengths}}{\end{list}}
\newcommand{\ListLengths}{\setlength{\itemsep}{0ex}\setlength{\topsep}{1ex}\setlength{\partopsep}{0ex}}

%% No indentation for enumeration
\newenvironment{NoIndentEnumerate}{\begin{list}{\arabic{listCounter}.}{
        \usecounter{listCounter}\setlength{\leftmargin}{1em}\ListLengths}}{\end{list}}


%% No indentation for the bullet.  The text is indented.  Nominal spacing.
\newenvironment{myitemize}{\begin{list}{$\bullet$}{\setlength{\leftmargin}{1em}}}{\end{list}}

%% No indentation for the bullet.  The text is indented.  Little spacing.
\newenvironment{mylowitemize}{\begin{list}{$\bullet$}{\setlength{\leftmargin}{1em}\ListLengths}}{\end{list}}

\newcommand{\PLS}{${\textbf{PLS}}$}
\newcommand{\PPAD}{${\textbf{PPAD}}$}
\newcommand{\Poly}{${\textbf{P}}$}
\newcommand{\TFNP}{${\textbf{TFNP}}$}
\newcommand{\FP}{${\textbf{FP}}$}
\newcommand{\NP}{${\textbf{NP}}$}
\newcommand{\RP}{${\textbf{RP}}$}
\newcommand{\Stwo}{${\textbf{S}^2_P}$}
\newcommand{\reduce}{$\leq_P$}

\newcommand{\leqprefer}{\preceq}
\newcommand{\geqprefer}{\succeq}
\newcommand{\ltprefer}{\prec}
\newcommand{\gtprefer}{\succ}
\newcommand{\eqprefer}{=}

\newcommand{\pInstance}{$\textbf{P}$}
\newcommand{\bInstance}{$\textbf{B}$}
\newcommand{\mInstance}{$\textbf{M}$}
\newcommand{\ib}{i}
\newcommand{\jb}{j}

% preference game reductions
\newcommand{\inedges}{\mbox{in}}
\newcommand{\outedges}{\mbox{out}}


\newcommand{\eol}{{\scshape{End of the Line}}}
\newcommand{\twonash}{{\scshape{2-NASH}}}
\newcommand{\scarf}{{\scshape{Scarf}}}
\newcommand{\core}{{\scshape{Core-Balanced-NTU}}}
\newcommand{\kernel}{{\scshape{Strong Kernel}}}
\newcommand{\hypergraph}{{\scshape{Fractional Hypergraph Matching}}}
\newcommand{\necklace}{{\scshape{Necklace Splitting}}}
\newcommand{\hamsandwich}{{\scshape{Ham Sandwich}}}
\newcommand{\stochastic}{{\scshape{Stochastic Game}}}
\newcommand{\threesat}{{\scshape{3-SAT}}}
\newcommand{\prefGame}{{\scshape{Preference Game}}}
\newcommand{\approxPrefGame}{{\scshape{$\epsilon$-Approximate Preference Game}}}
\newcommand{\constDegreePrefGame}[1]{{\scshape{Degree $#1$ Preference Game}}}
\newcommand{\threeDBrouwer}{{\scshape{3-D Brouwer}}}
\newcommand{\brouwer}{{\scshape{Brouwer}}}
\newcommand{\pe}{{\scshape{Personalized Equilibrium}}}
\newcommand{\constPlayersPe}[1]{{\scshape{$#1$-Personalized Equilibrium}}}
\newcommand{\constDegreePe}[1]{{\scshape{$#1$-Graphical Personalized Equilibrium}}}
\newcommand{\approxPe}{{\scshape{$\epsilon$-Approximate Personalized Equilibrium}}}
\newcommand{\spp}{{\scshape{SPP}}}
\newcommand{\fspp}{{\scshape{FSPP}}}
\newcommand{\bbc}{{\scshape{BBC}}}
\newcommand{\fbbc}{{\scshape{FBBC}}}
\newcommand{\plcp}{{\scshape{P-LCP}}}













